mixing flow characteristics in a vessel agitated by the

1

bpgstfiTbmafl

tTwrhfhtcs

cistwpism

JsY

J

Downloaded Fr

Yeng-Yung Tsui1

Professore-mail: [email protected]

Yu-Chang HuGraduate Student

Department of Mechanical Engineering,National Chiao Tung University,

Hsinchu 300, Taiwan

Mixing Flow Characteristics in aVessel Agitated by the ScrewImpeller With a Draught TubeThe circulating flow in a vessel induced by rotating impellers has drawn a lot of interestsin industries for mixing different fluids. It used to rely on experiments to correlate theperformance with system parameters because of the theoretical difficulty to analyze sucha complex flow. The recent development of computational methods makes it possible toobtain the entire flow field via solving the Navier–Stokes equations. In this study, acomputational procedure, based on multiple frames of reference and unstructured gridmethodology, was used to investigate the flow in a vessel stirred by a screw impellerrotating in a draught tube. The performance of the mixer was characterized by circula-tion number, power number, and nondimensionalized mixing energy. The effects on thesedimensionless parameters were examined by varying the settings of tank diameter, shaftdiameter, screw pitch, and the clearance between the impeller and the draught tube. Alsoinvestigated was the flow system without the draught tube. The flow mechanisms to causethese effects were delineated in detail. �DOI: 10.1115/1.2903815�

Introduction

The process of fluid mixing is commonly used in industries. Tolend different fluids, a vessel is equipped with a rotational im-eller. A swirling flow, i.e., a flow rotating about the vessel axis, isenerated by the impeller. The function of the impeller is to en-ure transport of a fluid element to anywhere in the tank. In ordero fulfill this objective, axial vortices, i.e., loop flows circulatingrom the top region of the tank to the bottom region, must benduced. Thus, the flow in the tank is inevitably three dimensional.he contact surfaces between different fluids are then deformedy the three-dimensional flow. Since different fluid materials areixed in the interface layer via molecular diffusion, the deformed

nd, thus, elongated contact surface leads to the enhancement ofuid mixing.For low-viscosity fluids, the most commonly used impellers are

he disk-type flat blade turbines and the pitched-blade turbines �1�.he low viscosity of the fluids and the smaller size, comparedith the tank, render the turbine impellers suitable for high-speed

otation, which makes the flow field turbulent. Turbulent flows areelpful in fluid mixing because the contact surfaces between dif-erent fluids are greatly wrinkled by the turbulence fluctuations. Atigher viscosities, these impellers lose their effectiveness due tohe difficulty to achieve high shear conditions because large poweronsumption is demanded. Instead, helical ribbon impellers orcrew impellers, rotating at low speeds, are preferred.

The screw impeller, despite its small diameter, produces signifi-ant secondary circulation when incorporating a draught tube ands the main concern of the present study. This mixer works in aimilar manner to the screw extruder. However, the former func-ions to maximize pumping flow with a low pressure increase,hereas the latter works to maximize pressure rise with a lowumping capacity. In the past, most analyses of the pumping flown the screw mixer are based on an analogy to the extruder. Aseen in Refs. �2,3�, the flow in a tube pumped by a screw can beainly divided into a drag flow qd and a pressure flow qp,

1Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

OURNAL OF FLUIDS ENGINEERING. Manuscript received January 31, 2007; final manu-cript received October 18, 2007; published online April 3, 2008. Assoc. Editor:r
u-Tai Lee.
ournal of Fluids Engineering Copyright © 20

om: http://fluidsengineering.asmedigitalcollection.asme.org/ on 04/25/201

q = qd − qp �1�

The drag flow qd resulting from the relative motion between thescrew and the casing is a forward flow in the positive direction ofthe axis. It was treated as a fully developed flow between twoparallel plates with one plate moving at the rotational speed todrive the fluid flow, which is similar to the well-known Couetteflow. When the screw pump operates, pressures builds up in thepump, resulting in a high pressure at the outlet. The adverse pres-sure gradient brings about a back flow in the negative direction ofthe axis. This pressure flow qp can be analyzed by assuming aPoiseuille flow between the plates.

The above simple models were also used to estimate powerrequirements. The power consumed by the drag flow is obtainedvia calculating the internal heat generation due to viscous dissipa-tion. Both the flows in the channel direction and in the transversedirection need to be taken into consideration in estimating theenergy dissipation. For the pressure flow, the power required is thepower expended as flow energy in raising the pressure of the fluid,which is equal to the product of the pressure rise in the screwpump and the maximum pumping flow rate corresponding to zeropressure gradients.

For the pressure flow, the pressure rise between the inlet and theoutlet of the screw pump needs to be known. In general, the pres-sure rise is determined from experiments. However, Sykora �4�used the frictional loss in the flow through the annular span be-tween the draught tube and the tank to estimate this pressure rise.To account for the effects of the side walls and the curvature ofthe flow channel in the pump at low helical angles, shape factorswere introduced �2,3�.

In the past, most studies were done to correlate three key char-acteristic parameters �power number, mixing time, and circulationnumber� to geometric parameters, such as the tank diameter, thelength of the screw, the pitch of the impeller, and the width of theimpeller blade �4–10�. These correlations were expressed in di-mensionless form to generalize their applicability. When geomet-ric configurations are fixed, the dependence of the three charac-teristic parameters on the rotational speed of the screw impeller�n� or the Reynolds number �Re� can be summarized for Newton-ian fluids as

NP Re = const �2�

APRIL 2008, Vol. 130 / 041103-108 by ASME

4 Terms of Use: http://asme.org/terms

wthvb

Tnvtl

rsettatpAnpsdrocpmmfitwem

2

tcoirrptqct

T

0

Downloaded Fr

NQ = const �3�

�mn = const �4�

here NP is the power number, NQ the circulation number, and �mhe mixing time. These relationships are valid for Re�20. Forigher Reynolds numbers, the results of Carreau et al. �11� re-ealed that mixing time gradually decreases with Reynolds num-er. They also found that

Np Re

NQ= const �5�

he circulation number increases from the value at low Reynoldsumbers with increasing Re to eventually attain another constantalue. It was concluded by them that the ratio of the mixing timeo the circulation time �the time for the flow to complete a circu-ation loop in the vessel� is a constant.

In view of the above review, despite the fruitful achievement inelating the screw mixer performance to the system configurationettings, details about the flow field were not delineated in thesexperimental works and the simple analytical models cannot por-ray the flow field precisely. Mathematical methods, which solvehe Navier–Stokes equations, give complete pictures of velocitynd pressure fields and are thus useful to analyze the flow. Al-hough they have already been widely employed in industrial ap-lications, their use in simulating the screw mixer is very limited.s far as we are aware, only Kuncewicz et al. �12� adopted aumerical method to study the effects of the tank diameter and theitch of the screw impeller on the mixer performance. In theirimulations, the flow field was simplified by assuming it to be twoimensional. The action of the rotating impeller on the flow isealized by imposing a form drag force and a frictional drag forcento the momentum equations �13�. Two empirical coefficients,haracterizing the two forces, need to be specified. Recently, theresent authors developed a three-dimensional computationalethod, which is based on a fully conservative finite volumeethod and incorporates unstructured grid techniques. Multiple

rames of reference were employed to tackle the rotation of thempeller. This method had been successfully used in the simula-ion of the vacuum pump of the Holweck type �14� and the mixerith pitched-blade turbines �15�. In this study, it is employed to

xamine the effects of various geometric parameters on the screwixer with a draught tube.

Mathematical MethodA schematic sketch of an agitated system with a screw impeller

ogether with a draught tube is shown in Fig. 1. For numericalalculations, the draught tube is divided into an inner part and anuter part. The outer part covers the clearance region between thempeller blade and the inner wall of the tube. In this clearanceegion and the bulk region outside the draught tube, the frame ofeference is stationary. A rotational frame is imposed on the innerart of the tube to allow the blade of the impeller to sweep acrosshis region. The flow field in the agitated vessel is assumed to beuasisteady, implying that the screw impeller is frozen at a spe-ific position. The governing equations in the rotational frame canhen be cast into the following form:

��Uj* − U

gj* ��

�xj* = 0 �6�

��Uj* − U

gj* ��U

i* − U

gi* ��

�xj* = −

�p*

�xi*

+1

Re

�

�xj*� ��U

i* − U

gi* �

�xj* �

− �mni�mUgn* − 2�mni�m�U

n* − U

gn* �

�7�
he above equations are expressed in dimensionless form in
41103-2 / Vol. 130, APRIL 2008


which the coordinates, velocities, and pressure are normalized as

xj* =

xj

d, U

j* =

Uj

nd, p* =

p

�n2d2 �8�

where d is the diameter of the impeller, n the rotational frequency,and � the density of the fluid. The Reynolds number Re is definedas �nd2 /�, where � is the molecular viscosity of the fluid. U

j*

−Ugj* represents the flow velocity with respect to the moving grid

and Ugj* =� jpq�px

q* is the velocity of the moving grid rotating with

the impeller. � j�=2�nej� is the angular velocity of the impeller.The effects of the rotational frame are represented by the bodyforces given in the last two terms in the momentum equation,corresponding to the centrifugal force and the Coriolis force. Inthe stationary frame of reference, the grid velocity U

gj* =0 and the

body forces are dropped.There is a need to have a comprehensive tool to handle various

flow configurations encountered in industries. To meet this re-quirement, the authors have developed a general code in recentyears such that the flow with irregular boundaries can be ana-lyzed. The analytic tool is based on a fully conservative finitevolume method applicable to unstructured grids, which are madeof polyhedrons with arbitrary geometric topology. This method isbriefly described in the following.

The transport equations are first integrated over a control vol-ume which, in principle, can be of arbitrary geometry. With theaid of the divergence theorem, the volume integrals of the con-vection and diffusion terms are transformed into surface integrals.It is followed by the use of midpoint rule to yield discretizationform

�f

��U� − U� g� f · s� f�� f = �f

1

Re� � f · s� f + qv �9�

where the subscript f denotes the face values, s� f is the surfacevector of the considered face, and � represents the components of

the relative velocity �U� −U� g�. The last term in the equation repre-sents all the terms apart from the convection and diffusion terms.

Fig. 1 A sketch of the mixing system

The summation is taken over all the faces of the control surface.

Transactions of the ASME


a

w

uvI0dI

m

w

td�n

twmbfeevt

macTeatmR

3

ithD=l−

Fw

J

Downloaded Fr

The convective value � f through each face is approximated byhigh-order scheme,

� f = �UD + ��UD · �� 10�here the superscript UD denotes the value evaluated at a node

pstream of the face under consideration and �� is the distanceector directed from the upwind node to the centroid of the face.n the equation, is a blending factor falling in the range between

and 1. For a value of 0, it simply degenerates into the upwindifference scheme. For =1, the scheme is second order accurate.n the present calculations, a value of 0.9 is assigned to .

The diffusive flux is further modeled by the following approxi-ation:

1

Re� � f · s� f =

1

Re

sf2

��PC · s� f

��C − �P� +1

Re�� f · �s� f −

sf2

��PC · s� f

��PC��11�

here �see Fig. 2� the subscripts P and C denote the principal and

he neighboring nodes sharing a common face f , and ��PC is aistance vector connecting these two nodes. The face gradient� f is obtained via interpolation from the gradients at the twoodes.

In solving the discretized transport equations, the first terms onhe right-hand side in Eqs. �10� and �11� are implicitly treated,hile the second terms in the equations are tackled in an explicitanner using the deferred-correction procedure. It also needs to

e noted that the computational molecules for the nodes in onerame immediately adjacent to the interface between the two ref-rence frames include nodal points in the other frame. To correctlyvaluate the momentum flux transported through the interface, theelocities at these neighboring nodes must be transformed ontohe frame where the considered node is located.

After solving momentum equations, the resulting velocitiesust be adjusted, and the prevailing pressure must be upgraded inway that the continuity equation is satisfied. The enforcement of

onservation of mass results in a pressure-correction equation.he discretized momentum equations and the pressure-correctionquation are sequentially solved in an iterative manner. Iterationsre performed to account for the nonlinearities, the coupling be-ween velocities and pressure, and the deferred-correction terms

entioned above. More details about the method can be found inefs. �14,16�.

Results and DiscussionA drawing of the mixing system with a screw-in-a-tube agitator

s given in Fig. 1. A benchmark configuration of the mixing sys-em has a screw impeller with diameter d=126 mm and length=1.5d. The diameter D and the height H of the agitated tank are=H=2.3d. The screw impeller has a shaft of diameter dS0.254d and a clearance C=0.05d. The pitch of the screw impel-

er is S=0.6d. The width of the screw blade W is equal to �d

ig. 2 Illustration of a principal node and a neighboring nodeith a common face

dS� /2, and the distances between the impeller and both the top

ournal of Fluids Engineering


and the bottom of the tank are h1= �H−h� /2. The impeller rotatesat a speed of 59 rpm, which is equivalent to a Reynolds number of1.52.

To generate unstructured grids, the computational domain isfirst divided into 80 blocks. In each block, an algebraic method isused to create a suitable grid. After the grids for all the blocks areconstructed, the grid nodes are readdressed. An example of theresulting mesh is presented in Fig. 3. To examine grid effects onthe mixing flow performance, grids with about 120,000, 200,000,300,000, and 380,000 cells have been tested. It was found that thepower number �defined by Eq. �14�� gradually increases with thecell number from 287 to 301.6. However, the difference betweenthe two finest grids is only 0.4%. The circulation number �definedby Eq. �12�� is less sensitive to the choice of grid. It varies from0.196 for the coarsest grid to 0.193 for the other grids. In thefollowing, the grid with about 380,000 cells is adopted for calcu-lations. The no-slip conditions are imposed on all the surfaces ofthe system. The flow in the vessel is assumed to be motionless asthe initial condition.

The flow field, illustrated in Fig. 4, shows that fluid is drawn bythe rotating impeller into the draught tube from the upper openingand emerges from the bottom end, followed by an upward flowoutside the tube to complete a circulation loop. To validate themethodology, calculations have been conducted to compare withthe measurements of Seichter et al. �8� and Seichter �9�. In theexperiments, the liquids to be mixed were solutions of starchsyrup in water, which exhibit a Newtonian behavior. The flowvelocities were measured using a thermal resistor. To measure thetorque, the mixing vessel was placed on a turntable. The torqueproduced by the rotating impeller was transmitted to a silon threadand compensated on a desk balance. There are two major differ-ences between the experiments and the simulations. One is thatthe thickness of the impeller is not given in the experiments.Therefore, a guessed value for the thickness was used in simula-tions. The other is the upper surface of the flow in the vessel,which is open to air. However, it is assumed to be a solid wall in

Fig. 3 A computational mesh

calculations. A comparison of the axial velocity in the annular

APRIL 2008, Vol. 130 / 041103-3


rT1b5ffltvpws

Fi

0

Downloaded Fr

egion outside the tube at the midheight plane is shown in Fig. 5.hree rotational speeds, corresponding to Re=0.92, 1.26, and.52, were under consideration. Since the thickness of the impellerlade was not clear, three values, corresponding to t=1 mm,mm, and 8 mm, were chosen to test its effects. It is obvious

rom the figures that the larger the blade thickness, the lower theow rate. Both the experiments and the predictions indicate that

he flow in the annulus approaches a fully developed state with theelocity close to a parabolic profile. A further comparison betweenredictions and measurements for Re=1.52 is given in Table 1 inhich the circulation number NQ and the power number N

P* are

hown. The corresponding NQ and NP* for Re=1.26 and 0.92 are

Fig. 4 Typical flow field

ig. 5 Comparison of predicted and measured axial velocities
n the annular region outside the tube
41103-4 / Vol. 130, APRIL 2008


almost the same as the values listed in Table 1 due to the factthese numbers are independent of the Reynolds number, as seenfrom Eqs. �2� and �3�. For the largest thickness case, the predictedNQ is close to the experimental data, but the predicted N

P* is too

high. For t=5 mm, the predicted NP* is in good agreement with the

measurements and the predicted NQ is only about 8% higher.Therefore, 5 mm of blade thickness, being about 4% of the im-peller diameter, is used throughout the following results.

The performance of the mixer is characterized in terms of threedimensionless parameters: circulation number NQ, power numberN

P*, and energy of mixing E.The circulation number is defined by

NQ =Q

nd3 �12�

Here, Q is the volumetric flow rate through the draught tube. It isexactly the same as the flow rate in the annulus outside the tubedue to mass conservation.

The power number is usually given as

NP =P

�n3d5 �13�

where P is the power consumed by the impeller. The power re-quired to drive the impeller can be divided into two parts: one dueto the pressure force acting on the impeller blade and the otherdue to the frictional force acting on the surfaces of the blade andthe shaft. This power number is a function of Reynolds number.Another form of the power number can be expressed by

NP* =

P

�n2d3 �14�

These two expressions are related by

NP* = NP Re �15�

As seen from Eq. �2�, NP* is a constant for low Reynolds numbers.

A criterion to evaluate the effectiveness of the mixer is theenergy of mixing E nondimensionalized as �12�

E =P�m

�nd3 �16�

where �m is the time needed for the mixing process to reach ahomogenization state. To estimate the mixing time, unsteady flowsimulations must be undertaken, which require large computa-tional time. Alternatively, we can use the circulation time insteadof the mixing time because it was observed that the time requiredfor mixing is proportional to the time required for circulation �11�.The circulation time is given by

�c =V

Q�17�

where V is the volume of the considered tank. By substituting

Table 1 Comparison between predicted and measured circu-lation numbers and power numbers for different bladethicknesses

t=1 mm t=5 mm t=8 mm Expt.

NQ 0.204 0.193 0.183 0.178

NP* 242.6 301.6 341.6 306.1

4�c into Eq. �16� to replace �m �11�, we can redefine E as



Iqfft

ttppal�afiei�ntcrtiNthWc2t

aqtAf6aaomup

ttNfnftiofcctscmIta

J

Downloaded Fr

E = �N

P*

NQ

�H

D��D

d�3

�18�

t is noted that in the equation, NP* /NQ represents the power re-

uired per unit of circulating flow. It can be used as an indicatoror pumping effectiveness �15� but is not suitable for mixing ef-ectiveness because the geometric size of the tank needs to beaken into account in the mixing process.

3.1 Effects of Tank Diameter. To examine the influence ofhe relative size of the stirred tank to the impeller, the diameter ofhe tank is gradually enlarged, while those of both the screw im-eller and the draught tube remain unchanged. Its effects on theumping capacity, power consumption, and mixing effectivenessre illustrated in Figs. 6�a�–6�c�. In the figures, two sets of calcu-ations corresponding to cases with a constant impeller shaftds /d=0.254, where d is fixed� and a variable shaft �ds /D=0.11�re presented. In the constant shaft case, the pump configuration isxed, while the pump shaft diameter ds will be increased with thenlarged tank in the variable shaft case. The inclusion of the latters simply used to compare with the experiments of Seichter et al.8� and the predictions of Kuncewicz et al. �12� because there areo data available for the former case for comparison. Consideringhe case with a fixed shaft, for small values of D close to d, theirculating flow rate NQ is small because of the great frictionalesistance of the narrow passage between the draught tube and theank wall. The flow rate quickly builds up when the annular spaces enlarged to D /d=2.3, followed by leveling off to a value of

Q=0.35. Theoretically, the power number NP* is maximized as

he tank diameter is reduced to the tube diameter because a veryigh pressure gradient is generated, but without any fluid flow.hen the tank diameter is increased, the power requirement de-

reases sharply and then NP* reaches nearly a constant value of

25 after D becomes larger than 2.3d. Also shown in Fig. 6�b� arehe two parts of power number due to the pressure force �N

P,p* �

nd the frictional force �NP,f* �. For small tank diameters, N

P,p*

uickly decreases and NP,f* gradually increases. Both are attributed

o the release of the flow in the annular region as D /d increases.t large diameters, power consumption is mainly attributed to the

rictional loss, which is about three times the pressure loss. Figure�c� illustrates that the energy required for mixing is reduced firstnd then quickly increases when the tank is enlarged. There existsminimum point at D /d=2. This value is smaller than the value

f 2.3 at which the circulating flow rate becomes nearly a maxi-um and the power consumption a minimum. The results can be

nderstood in view of Eq. �18� in which the mixing energy isroportional to D3.For the variable shaft case with ds /D=0.11, the flow channel in

he draught tube is narrowed as the shaft diameter increases withhe enlarged tank, resulting in a reduction in NQ and an increase in

P* when D is greater than 2.3d. A similar trend for NQ can be

ound in the results of Kuncewicz et al. �12�. However, the powerumber N

P* increases with increasing D in the present calculations

or D greater than 2.3d but continues to decrease in the calcula-ions of Kuncewicz et al. It will be shown in Sec. 3.2 that anncrease in shaft diameter will cause an increase in frictional forcen the shaft surface, which leads to a large frictional loss. There-ore, the predictions of Kuncewicz et al. are not reliable. Theause of the decreasing behavior is thought to be due to theiralculation procedure. As mentioned in the Introduction, in theirwo-dimensional calculations, two drag coefficients need to bepecified to account for the action forces by the impeller. Theseoefficients were determined by some pilot tests to fit the experi-ental data and then used throughout the rest of the calculations.

t is conjectured that these coefficients may require adjustment forhe cases with large D /d. This may explain why their results havegood agreement with the measurements at low values of D /d.
The experimental data of Seichter et al. �8� are also included in


the figures. There exist some differences between the present

Fig. 6 Variation of „a… circulation number, „b… power number,and „c… mixing energy against tank diameter D

predications and the measurements. It needs to be emphasized that

APRIL 2008, Vol. 130 / 041103-5


tcd

Fa

0

Downloaded Fr

he simulation cannot completely mimic the experiments. As dis-ussed above, the thickness of the impeller blade and that of theraught tube were not clear in the experiments.

ig. 7 Variation of „a… circulation number, „b… power number,nd „c… mixing energy against shaft diameter ds

3.2 Effects of Shaft Diameter. As seen in Fig. 7�a�, the cir-

41103-6 / Vol. 130, APRIL 2008


culating flow rate is reduced by enlarging the shaft. This is notunexpected because the height and, thus, the cross-sectional areaof the flow channel in the tube decrease. The power number,shown in Fig. 7�b�, remains almost a constant until ds /d=0.6,followed by an increase. It is seen in the figure that the increase inthe power number at large shaft diameters is due to the increase infrictional loss �N

P,f* �. The enlarged shaft causes an increase in the

surface velocity and a reduction in the channel height. As a con-sequence, the shear stress in the channel is increased, which isevidenced in view of the radial variation of circumferential veloc-ity V� shown in Fig. 8. Since the shaft surface area also increaseswith the shaft diameter, the frictional loss is greatly increased atlarge diameters. The required mixing energy, as shown in Fig.7�c�, gradually increases for low values of dS until dS /d=0.6. It isthen followed by a rapid increase.

3.3 Effects of Impeller Pitch. A schematic drawing of theflow channel formed by the impeller blade and the draught tube isdeployed onto a plane, as shown in Fig. 9, in which the clearancebetween the impeller and the tube is neglected. The length of thechannel for one revolution of impeller blade is given by L=S /sin , where is the helical angle. The flow in the channelcan be regarded as being driven by the internal wall of the tubemoving in a direction opposite to that of the rotation. This movingwall velocity �Vw� can be divided into two components: one in thechannel direction �Vw cos � and the other in the transverse direc-

Fig. 8 Distribution of circumferential velocity along the radialdirection in the draught tube for different shaft diameters

Fig. 9 A schematic drawing of the flow channel inside the
draught tube


tbbdf

waAawaetddabtteeNe

flicnHi= tibpiqvft

t

t

Fd

J

Downloaded Fr

ion �Vw sin �. A pressure rise between the inlet and the outlet isuilt up by the channel-direction flow and a pressure differenceetween the side walls by the transverse flow. These two pressureifferences are related by considering the axial component of theorce balance over the entire channel,

�pout − pin�Az = �pp − ps�As cos − Fz �19�

here pin and pout are the pressures at the inlet and the outlet, ppnd ps are the pressures at the pressure side and the suction side,z is the cross-sectional area, As is the sidewall area, and Fz is thexial component of the frictional force exerted on the channelalls. In the equation, the momentum difference between the inlet

nd the outlet is neglected. It is obvious that the pressure differ-nce between the sidewalls is linearly related to the pressure risehrough the draught tube. As an illustration, the pressure contoursistributed on the impeller shaft are presented in Fig. 10. Theimensionless average pressures at the inlet and exit of the tubere pin=91.6 and pout=100.8, while those on the two sides of thelade are pp=104 and ps=87.9. As a result, the pressure risehrough the tube is pin− pout=9.2 and the pressure difference be-ween the two blade sides is pp− ps=16.1. It is noted that therexists a high pressure zone at the leading edge of the blade at thentrance and a low pressure zone at the trailing edge at the exit.ear these two regions, reversed flows may result. However, its

ffect on the primary flow structure is insignificant.The pitch, or the helical angle , has two opposite effects on the

ow. As seen in Fig. 9, by decreasing the helical angle, the veloc-ty component of the moving wall in the channel direction is in-reased, being helpful to the fluid flow. However, the channel isarrowed and lengthened, resulting in greater frictional resistance.ence, there exists an optimum pitch value for maximum flow. It

s observed in Fig. 11�a� that this optimum value holds at S /d1.5 in the present computations. For the two limiting cases with=0 deg and 90 deg, i.e., S /d=0 and �, there is no flow through

he draught tube because in the former the impeller degeneratesnto a rotating cylinder, whereas in the latter the impeller bladeecomes vertical. For the rotating cylinder case, a large amount ofower is required to overcome the great frictional resistance thats prevailing. As shown in Fig. 11�b�, the power requirementuickly decreases first, followed by leveling off to a constantalue. When S is greater than 1.5d, the constant value of N

P* is 226

or S /d=�. It is also shown in the figure that the pressure part ofhe power number N

P,p* gradually approaches 226 while the fric-

ional part NP,f* gradually approaches zero. For the limiting case of

ig. 10 Pressure distribution on the impeller shaft from twoifferent view angles

he vertical blade, transverse flow prevails in the draught tube,



which generates large pressure difference between the two side-walls. The variation of mixing energy in Fig. 11�c� shows that the

Fig. 11 Variation of „a… circulation number, „b… power number,and „c… mixing energy against screw impeller pitch S

most efficient mixing occurs when S /d=1.5.

APRIL 2008, Vol. 130 / 041103-7


aKlouat

stfieclitnc1ss

oimictdtdsrcFia=cepvdtittvdrldwbtt

wcbTTiwg

0

Downloaded Fr

Compared with the measurements of Seichter et al. �8�, thegreement is reasonably good. However, the calculations ofuncewicz et al. �12� showed that both NQ and N

P* are much

arger than the present predictions for high values of S. Theseverpredictions are believed to be due to the drag coefficientssed in their model. However, despite the large differences in NQnd N

P*, the two calculations yield a close agreement in predicting

he mixing energy E.

3.4 Effects of Clearance Gap. It was discussed that a pres-ure difference is generated between the two sidewalls by theransverse flow, as seen in Fig. 9. The sidewalls represent the twoaces of the impeller blade. In practical applications, a passage,.e., a clearance gap, exists to connect the two sides. Due to thexisting pressure gradient, part of the fluid flows through thelearance from the pressure side to the suction side. This floweakage causes a decrease in the sidewall pressure difference and,n turn, a decrease in pressure rise in the flow channel becausehey are related, as seen in Eq. �19�. Consequently, the circulationumber NQ, the power number N

P*, and the mixing energy E de-

rease with increasing clearance C, which are shown in Figs.2�a�–12�c�. It is seen from Fig. 12�b� that the clearance has aignificant effect on the frictional force and, thus, the power con-umption when the clearance is very small.

3.5 Effects of Draught Tube. A comparison has been maden a system with and without a draught tube by gradually increas-ng the tank size. It is known that without a tube, a surface vortex

ay be formed due to the swirling flow generated by the rotatingmpeller. This surface vortex flow does not exist in the presentalculations by assuming that the upper surface is flat. Accordingo Fig. 13�a�, the circulation number NQ for the case without araught tube is almost linearly related to the tank diameter D inhe considered range. For low values of D, the system with araught tube is superior to that without one. However, it losesuperiority when D�4d. For the sake of better understanding thisesult, the mean axial velocity profiles �averaged over the entireircumference� for D /d=5.0 at midheight of the tank are shown inig. 14. It is seen that by removing the tube, the downward flow

nduced by the impeller continuously extends to the region farway from the impeller, the point at which the axial velocity Vz0 is the center of the circulation loop. The variation of thisirculation center with the tank diameter is shown in Fig. 15. It isvident that the location of this circulation center is nearly pro-ortional to the tank diameter. This explains the previous obser-ation that the flow number NQ is linearly related to D when theraught tube is not incorporated. It is interesting to notice that forhe large tank with D=5.0d shown in Fig. 14, a small amount ofnduced circulating fluid between the draught tube and the wall ofhe tank bypasses the tube and flows over the outside surface ofhe tube. As an illustration of the flow field, the streamlines on aertical plane for the D /d=5 case is prepared in Fig. 16. With theraught tube, there exist small vortices near the inlet and the outletegions just outside the tube together with the primary circulatingoop flow. By removing the tube, the above observed vorticesisappear. However, owing to the reduced strength of the down-ard pumping flow and the enlarged space, a vortex is formedeneath the impeller. It can be seen that without the restriction ofhe tube, the center of the induced looping flow moves away fromhe impeller, as discussed in the above.

The power number, shown in Fig. 13�b�, is nearly a constanthen the tank diameter D becomes greater than 2.5d. The power

onsumption is much larger for the case with a draught tube,eing about 2.2 times greater than that without a draught tube.his result is supported by the experiments of Chavan et al. �5�.he lower power consumption in the case without a draught tube

s mainly ascribed to the large space between the impeller and theall of the tank, which, in turn, leads to a reduction of pressure
radient and, thus, the power required, as discussed in Sec. 3.4.
41103-8 / Vol. 130, APRIL 2008


The mixing energy required, shown in Fig. 13�c�, for the case witha draught tube is higher than that without a tube. The differencebecomes very significant for large tanks.

4 ConclusionsA three-dimensional computational method has been employed

to investigate the mixing flow stirred by a screw impeller with adraught tube. The main findings are summarized in the following.

�1� When the tank diameter is slightly greater than that of thedraught tube, the narrow passage in the annular space out-

Fig. 12 Variation of „a… circulation number, „b… power number,and „c… mixing energy against clearance C

side the tube restricts the fluid flow and causes large power



Fai

J

Downloaded Fr

consumption. Either the circulation number or the powernumber nearly reaches a constant for D�2.3d. However, interms of the total energy required for mixing, the optimumsize of the tank occurs at D=2d.

ig. 13 Variation of „a… circulation number, „b… power number,nd „c… mixing energy against tank diameter D for the screw

mpellers with and without a draught tube

�2� The increase in the shaft diameter leads to a decrease in



channel height and, thus, reduction in fluid flow. The powerrequirement remains nearly a constant until ds=0.6d. It isfollowed by an increase in power consumption due to theenhanced frictional force near the shaft surface.

�3� The flow inside the tube can be considered as a channel

Fig. 14 Distribution of axial velocity along the radial directionfor the screw impellers with and without a draught tube

Fig. 15 Variation of circulation center with tank diameter forthe case without a draught tube

Fig. 16 Flow streamlines on a vertical plane for the screw
mixer „a… with a draught tube and „b… without a draught tube
APRIL 2008, Vol. 130 / 041103-9


A

TM

N

0

Downloaded Fr

flow with a moving wall. When the impeller pitch de-creases, the velocity component of the moving wall alongthe channel is increased, but the channel is lengthened andnarrowed, resulting in larger frictional resistance. Thus,there exists an optimum value of pitch, which holds ats /d=1.5.

�4� The flow leakage from the pressure side to the suction sidein the clearance gap brings about a reduction in the pressuredifference between the two side surfaces of the impellerblade and, hence, reduces the pressure rise in the tube. As aconsequence, both the flow rate and the power requirementdecrease.

�5� Without a draught tube, the circulating flow rate is linearlyproportional to the size of the tank. Therefore, for suffi-ciently large tanks, the induced flow rate becomes greaterthan the system with a draught tube. The linear relationshipresulted from the radial movement of the center of the cir-culating flow. The power requirement for that without adraught tube is about half of that of the system with adraught tube. The lower power required is mainly due tothe lower pressure gradients caused by the open space be-tween the impeller and the wall of the tank.

cknowledgmentThis work was supported by the National Science Council of

aiwan, R.O.C. under the Contract No. NSC 96-2221-E-009-135-Y2.

omenclatureAs � sidewall area of pumping channelAz � cross-sectional area of pumping channelC � clearanced � impeller diameter

ds � shaft diameterD � tank diameterej � unit vector in the jth direction of Cartesian

coordinatesE � mixing energy �=��N

p* /NQ��H /D��D /d�3�

f � the faces of the control surfaceEz � axial component of the frictional force exerted

on pumping channel wallsh � impeller length

h1 � distance between the impeller and the top andbottom of the tank

H � liquid height in the tankL � length of the channel for one revolution of

impeller bladen � rotational frequency of the screw impeller

Np � power number �=P /�n3d5�N

p* � power number �=P /�n2d3�

Np,f* � power number due to frictional force

Np,p* � power number due to pressure force

NQ � circulation number �=Q /nd3�p � pressure

pin � pressure at the inlet of the pumping channelpout � pressure at the outlet of the pumping channel

pp � pressure at the pressure side of the pumpingchannel

ps � pressure at the suction side of the pumpingchannel

P � power consumption for a mixing vesselQ � volumetric flow rater � radial distance

Re � Reynolds number �=�nd2 /��

41103-10 / Vol. 130, APRIL 2008


S � screw pitchs f � surface vector of the considered facet � thickness of the screw blade

Ugj � grid velocityVw � moving wall velocityVz � axial velocityV� � circumferential velocityW � blade width

Greek Symbols � helical angle of the pumping channel � blending factor

�� distance vector directed from the upwind nodeto the centroid of the face

��PC � distance vector connecting the principal andthe neighboring nodes

� � fluid viscosity� � fluid density

�c � circulation time�m � mixing time� j � angular velocity of the screw impeller

SubscriptsC � neighboring nodef � face value

P � principal node

SuperscriptsUD � the value evaluated at a node upstream of the

face* � dimensionless variables

References�1� Tatterson, G. B., 1991, Fluid Mixing and Gas Dispersion in Agitated Tanks,

McGraw-Hill, New York.�2� Paton, J. B., Squires, P. H., Darnell, W. H., Cash, F. M., and Carley, J. F., 1959,

“Chap. 4 Extrusion,” Processing of Thermoplastic Materials, E. C. Bernhardt,ed., Van Nostrand Reinhold, New York.

�3� Carley, J. F., 1962, “Single-Screw Pumps for Polymer Melts,” Chem. Eng.Prog., 58�1�, pp. 53–58.

�4� Sykora, S., 1966, “Mixing of Highly Viscous Liquids,” Collect. Czech. Chem.Commun., 31, pp. 2664–2678.

�5� Chavan, V. V., Jhaveri, A. S., and Ulbrecht, J., 1972, “Power Consumption forMixing of Inelastic Non-Newtonian Fluids by Helical Screw Agitators,” Trans.Inst. Chem. Eng., 50, pp. 147–155.

�6� Chavan, V. V., and Ulbrecht, J., 1973, “Power Correlations for Close-Clearance Helical Impellers in Non-Newtonian Liquids,” Ind. Eng. Chem. Pro-cess Des. Dev., 12�4�, pp. 472–476.

�7� Chavan, V. V., and Ulbrecht, J., 1973, “Internal Circulation in Vessels Agitatedby Screw Impellers,” Chem. Eng. J., 6, pp. 213–223.

�8� Seichter, P., Dohnal, J., and Rieger, F., 1981, “Process Characteristics of ScrewImpellers With a Draught Tube for Newtonian Liquids: The Power Input,”Collect. Czech. Chem. Commun., 46, pp. 2007–2020.

�9� Seichter, P., 1981, “Process Characteristics of Screw Impellers with a DraughtTube for Newtonian Liquids: Pumping Capacity of the Impeller,” Collect.Czech. Chem. Commun., 46, pp. 2021–2031.

�10� Seichter, P., 1981, “Process Characteristics of Screw Impellers With a DraughtTube for Newtonian Liquids: Time of Homogenization,” Collect. Czech.Chem. Commun., 46, pp. 2032–2042.

�11� Carreau, P. J., Paris, J., and Guerin, P., 1992, “Mixing of Newtonian andNon-Newtonian Liquids: Screw Agitator and Draft Coil System,” Can. J.Chem. Eng., 70, pp. 1071–1082.

�12� Kuncewicz, C., Szulc, K., and Kurasinski, T., 2005, “Hydrodynamics of theTank With a Screw Impeller,” Chem. Eng. Process., 44, pp. 766–774.

�13� Kuncewicz, C., and Pietrzykowski, M., 2001, “Hydrodynamic Model of aMixing Vessel With Pitched-Blade Turbines,” Chem. Eng. Sci., 56, pp. 4659–4672.

�14� Tsui, Y.-Y., and Jung, S.-P., 2006, “Analysis of the Flow in Grooved PumpsWith Specified Pressure Boundary Conditions,” Vacuum, 81, pp. 401–410.

�15� Tsui, Y.-Y., Chou, J.-R., and Hu, Y.-C., 2006, “Blade Angle Effects on theFlow in a Tank Agitated by the Pitched-Blade Turbine,” ASME J. Fluids Eng.,128, pp. 774–782.

�16� Tsui, Y.-Y., and Pan, Y.-F., 2006, “A Pressure-Correction Method for Incom-pressible Flows Using Unstructured Meshes,” Numer. Heat Transfer, Part B,49, pp. 43–65.



mixing flow characteristics in a vessel agitated by the

Documents